积分符号 0到pai/2 dx/(cosx+2)

积分符号 0到pai/2 dx/(cosx+2)

∫[0,π/2] dx/(cosx+2) 令x=2t
= ∫[0,π/4] 2/(2+cos2t) dt
=∫[0,π/4] 2/ [1+2(cost)^2]dt 上下同除(cost)^2
=∫[0,π/4] 2(sect)^2 / [2+(sect)^2]dt
=∫[0,π/4] 2 / [2+(sect)^2] dtant
=∫[0,π/4] 2 / [3+(tan t)^2] dtan t
=(1/√3) ∫[0,π/4] 2 / [1+(tan t /√3)^2] d tan t /√3
=(2/√3) arctan( tan t / √3) |[0,π/4]
= π/(3√3)